[/quote]
,
and 
,
as the greatest exponent 
divides 
.
and infinitely many 
such that 
.
and 
.
with circumcircle 
is given. Let 
be the midpoint of 
,
be the foot of altitude from 
to 
,
be the intersection point of 
and 
.
lies on the arc 
.
.
be a trapezium inscribed in a circle 
with diameter 
. The circle with center 
and radius 
meets 
and 
(where 
)
at 
.
is perpendicular to 
.
![\[ x + y^2 = p^m \]](http://latex.artofproblemsolving.com/e/c/f/ecf553036f22327dd457dc06611b6bc1b0ef072c.png)
![\[ x^2 + y = p^n \]](http://latex.artofproblemsolving.com/5/b/d/5bd672934a395aabd6865ca4b50d8da75c0fe07e.png)
For 
positive integers and 
being a prime.
and 
.
Equality holds when 
or 
.
at 
intersect again at 
.
Equality holds when 
intersect its circumcircle 
.
determine a triangle 
,
is relative to 
are concurrent at a point 
and that 
lie in the interiors of segments 
respectively, such that 
Let 
intersect 
at 
,
intersect 
at 
.
and 
are either parallel or coincide.
and 
.
Where 
such that :
.
Y by Adventure10, Mango247
Two circles of radius 
and 
, with 
, are tangent to each other externally. The sides adjacent to the base of an isosceles triangle are common tangents to these circles. The base of the triangle is tangent to the circle of the greater radius. Determine the length of the base of the triangle.
and 
, with 
, are tangent to each other externally. The sides adjacent to the base of an isosceles triangle are common tangents to these circles. The base of the triangle is tangent to the circle of the greater radius. Determine the length of the base of the triangle.This post has been edited 1 time. Last edited by parmenides51, Apr 8, 2020, 11:41 PM
